Some improvements on the constants for the real Bohnenblust-Hille inequality

2010/09/14 by Daniel Pellegrino, Pellegrino, Daniel, Juan B. Seoane‐Sepúlveda +2
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical Inequalities and Applications #math.FA

paper · pdf · doi:10.48550/arxiv.1009.2717

9 pages. This note is an improvement of a previous version registered on arXiv

openalex publication_date 2010/09/14 · arxiv created 2010/10/04 · arxiv updated 2010/10/05 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

A classical inequality due to Bohnenblust and Hille states that for every N ∈ ℕ and every m-linear mapping U:ℓ_∞^N×...×ℓ_∞^N→ℂ we have \[(∑_i₁,...,i_m=1^N| U(e_i^₁,...,e_i_m)| ^(2m)/(m+1)) ^(m+1)/(2m)≤ C_m| U|] where C_m=2^(m-1)/(2). The result is also true for real Banach spaces. In this note we show that an adequate use of a recent new proof of Bohnenblust-Hille inequality, due to Defant, Popa and Schwarting, combined with the optimal constants of Khinchine's inequality (due to Haagerup) provides quite better estimates for the constants involved in the real Bohnenblust-Hille inequality. For instance, for 2≤ m≤ 14, we show that the constants C_m=2^(m-1)/(2) can be replaced by 2^\fracm²+6m-88m if m is even and by 2^\fracm²+6m-78m if m is odd, which substantially improve the known values of C_m. We also show that the new constants present a better asymptotic behavior.

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